Phaser-Based Transfer Function Analysis of Power Synchronization Control Instability for a Grid Forming Inverter in a Stiff Grid

The grid forming inverter (GFM) is considered a key technology to solve the issues of inverter-based resource (IBR)-rich grids. Many studies have investigated how GFM affects and improves power system stability. However, few studies have considered the instability of GFM as it relates to various grid conditions. This paper discusses the mechanism of power synchronization control instability of GFM by conducting transfer function analysis with a focus on various grid conditions, the control logic of GFM, and relevant parameter setting. The above analysis reveals that the instability of GFM is likely to occur when it is connected to stiff grids or has a larger phase delay due to a longer power measurement delay and lower damping of the controller. Based on these findings, a procedure for power synchronization controller design of GFM is proposed. Finally, experimental tests in the control hardware in the loop (CHIL) environment are performed to verify the validity of the transfer function analysis.


I. INTRODUCTION
Ggrowing awareness of climate change has led to the mass adoption of inverter-based resources (IBRs) such as solar photovoltaic generation (PV), wind power generation, and battery energy storage systems (BESSs) in power grids. With the expansive penetration of IBRs, a variety of grid issues have been observed, such as a higher rate of change of frequency (RoCoF) due to the reduction of system inertia, the instability of the phase-locked loop (PLL) on the grid following inverter (GFL) due to a lower short circuit ratio (SCR), and the lack of a short circuit current for operating the protection relay [1], [2], [3]. In these situations, the grid forming inverter (GFM) is considered a key technology to solve the challenges of IBR-rich grids [4], [5], [6], [7], [8].
Several papers apply control theory to power system stability analysis, including GFM [9], [10], [11]. GFM presents several implementations, but droop control-based GFM (GFM-Droop) and virtual synchronous generator The associate editor coordinating the review of this manuscript and approving it for publication was Tariq Masood .
(GFM-VSG) are typically regarded as the most popular methods [5]. Prior work [9] discusses how to design parameters equivalent to inertia constants and damping factor of VSG by deriving the transfer function of GFM in the phasor domain for the systems with GFM-VSG, but power measurement delay is not modeled therein. Other efforts [10] show the usefulness of the GFM by performing eigenvalue analysis of GFM-Droop and GFL with power measurement delay but neglect modeling of GFM-VSG with power measurement delay and instability of the GFM. Further, [11] derives transfer functions and state-space representations of GFM-Droop and GFM-VSG considering power measurement delay and discusses their respective properties, yet does not include discussion on the conditions under which GFM becomes unstable.
Moreover, since GFM is expected to be utilized in weak grids where GFL cannot operate stably, many previous studies have been conducted on weak grids [12], [13], [14], [15], [16]. On the other hand, there are few studies on GFM connected to stiff grids [17], [18], [19], [20], [21]. One study [17] conducted impedance-based stability analysis on GFM-Droop and revealed that GFM-Droop is likely to be unstable in stiff grids. Another study [18] notes that the phase delay due to an integration of system properties, which causes the instability of GFM-Droop in the stiff grid. Further work [19] presents a state-space representation of GFM-VSG, including the power measurement delay and its eigenvalue analysis, showing that the real part of the eigenvalues is positive in the stiff grid condition. Moreover, [20] mentions that the real part of the eigenvalues of GFM-Droop with an inner current loop and outer current loop increase in accordance with decreasing grid impedance. Finally, [21] proposes the method of decoupling the active power control loop and the reactive power control in stiff grids. However, there remains the need for a simple and quantitative discussion about the interaction mechanism between the GFM's power synchronization control and stiff grids, as well as a corresponding design policy for the power synchronization controller.
Based on the above studies, the main objective of this paper is to quantitatively and simply discuss the mechanisms of power synchronization control instability for a GFM in stiff grids. The contributions of this paper are highlighted below.
• Simply explain the mechanism of instability due to the interaction of GFM power synchronization control and the stiff grid by phasor-based transfer function analysis • Consider the effect of a grid disturbance, such as a phase jump and frequency sweep, on the system stability analysis The power synchronization controller determines the active power output characteristics of the GFM. The control block diagram of GFM-Droop is shown in Fig. 1a. The parameter K , called the droop gain, determines the relationship between the frequency of the GFM and the active power output of the GFM. The other method, GFM-VSG, emulates synchronous generator dynamics. Generally, GFM-VSG contains an inertia constant (M ), synthetic damping factor (D VSG ), and governer gain (K GOV ) as configurable parameters (Fig. 1b). D VSG and K GOV can be integrated into D, as shown in (1) because both manipulate P ref according to the deviation between the nominal frequency and the frequency of the GFM.
In addition to K , M , and D, this paper considers the time constant of a low-pass filter used for the power feedback measurement (T m ). Introducing a low-pass filter to inverter software is a common technique to enhance the robustness of feedback value measurement against ripple and noise on the measured signals. However, many previous studies have ignored this low-pass filter in GFM because of their simplifications of system formulation. This paper deploys single-loop control as a control system of GFM [22]. However, it should be noted that even if multiloop control is deployed as a control system of GFM, the current control loop and voltage control loop dynamics can be decoupled from power synchronization control as provided by proper control parameter design [23]. The delay time due to computational execution and PWM modulation is ignored because these delays generally are hundreds of times faster than the integral terms of the power synchronization control.

2) REACTIVE POWER CONTROL
Although proportional control is widely introduced in many studies using GFM as a reactive power controller, this paper adopts the PI controller shown in Fig. 2. Apart from a proportional controller, the PI controller offers the ability to adjust the reactive power output of GFM with the given target value regardless of the grid impedance. The effect of a reactive power controller on a power synchronization control loop is negligible in a stiff grid, as shown in the latter Section II-C.  Fig. 3 shows the system studied in this paper. In this system, GFM is connected to the infinite bus via an LCL filter (X LCL f ) with a grid reactance (X g ). In this system, the active and VOLUME 11, 2023  reactive power output of GFM (P out , Q out ) are represented as (2) and (3) with the parameters and variables defined in Table 1.

B. GRID DYNAMICS
Provided that the voltage output of the GFM and the voltage phasor differences between the GFM and infinite bus slightly change from the steady state as (4), the deviation of the active and reactive power output of the GFM can be represented as (5) and (6), respectively [9].
In the stiff grid, δ 0 is small enough to apply the approximation shown in (7). For instance, sin δ 0 and δ 0 of the system with X LCL f = 0.05 [24] and X g = 0.2, P out = 1.0, V i = 1.0, V g = 1.0 are derived to be 0.25000 and 0.25268, respectively. If the grid reactance of the system (X g ) is less than 0.2, (7) can be regarded as an approximation with good precision.
By deploying the derivation of Section II-A and the other considerations of this section, a linearized system representation of GFM is introduced in Fig. 4. C(s) in Fig. 4 the open-loop transfer function of power synchronization control (O P (s)) can be obtained as (8) [9] with the transfer function of the reactive power control system (G Q (s)) shown in Fig. 4.
The term G Q (s)/(1 + G Q (s)H (s)) represents the transfer function of the reactive power control closed loop. Normally, a reactive power controller is designed to prevent an overshoot of the reactive power output against the step command. The closed-loop system without overshoot shows a gain equal to or less than 1.0. Additionally, δ 0 can be regarded as less than 0.25 in the stiff grid with a grid reactance less than 0.2, as explained in Section II-B. With this assumption, the coupling term of the closed-loop reactive power control in the power synchronization control loop can be ignored, as shown in (9) and (10).

III. TRANSFER FUNCTION ANALYSIS ON ACTIVE POWER CONTROL LOOP
The closed-loop transfer function of the power synchronization control system (L P (s)) is represented as (13) by using Thus, the closed-loop transfer functions of the power synchronization control system with GFM-Droop (L PD (s)) and GFM-VSG (L PV (s)) are derived as (14) and (15), respectively.

B. STABILITY CRITERION OF DROOP AND VSG
The simplest method to evaluate the stability of the systems represented by (14) and (15) is to apply Hurvitz's stability criterion. Because (14) represents a second-order system, the system with GFM-Droop is stable if all the coefficients of the denominator in (14) are positive. As shown in Table 2, (14) is clearly stable independent of the parameter selection. Equation (15) represents the third-order system, which should satisfy the condition shown in (16) in addition to the condition of having positive coefficients of the denominator.
If T m ̸ = 0, (16) is transformed to (17), the following insights are derived by examining the parameters in (17).
• Greater X g improves the system stability • Less T m improves the system stability • Greater D improves the system stability • The effect of M on the system stability depends on the values of the other parameters In particular, D represents the critical parameter for the system stability because it affects the criterion (17) in square order.
Provided that GFM-VSG does not have the low-pass filter for the power measurement (T m = 0), (16) is transformed to (18). It is clear that without measurement delay, the GFM-VSG never becomes unstable because D and M are positive values.
In Section III-B, it is confirmed that Hurwitz's stability criterion can discriminate system stability under a given set of grid conditions and control parameters. Here, Bode plot analysis is performed to more quantitatively evaluate system stability. The standard parameters used are shown in Table 2. It should be noted that the values of K and D are carefully selected to present the same frequency droop characteristics in a steady state, as explained in a later Section IV-A.

1) COMPARISON OF DROOP AND VSG
Stability evaluation using the Bode plot is based on the openloop transfer functions shown in (11) and (12). The Bode plot in Fig. 6 shows the gain and phase of the systems with GFM-Droop and GFM-VSG. Because the system with GFM-Droop (11) is a second-order system, its maximum phase delay is 180 • . The maximum phase delay of the system with GFM-VSG (12), a third-order system, is 270 • .    Table 3 shows the gain and phase margins of the system. A negative phase margin indicates unstable status. According to Table 3, GFM-VSG shows less phase margin than GFM-Droop and shows higher instability induced by the phase delay of the system.

2) THE EFFECT OF SCR
SCR is widely known as an indicator of grid strength, and grids with SCRs less than 5 are regarded as weak grids [25]. In this paper, the grid SCR is equivalent to the inverse of the grid reactance and represented on a per unit basis(X g ), such that a higher SCR value indicates a stiffer grid. According to Fig. 7, it is confirmed that a higher SCR results in a higher system gain. Consequently, the GFM-VSG connected to the stiff grid contains a risk for instability, as shown in Table 4. This conclusion disputes the conventional understanding that IBRs are stable in stiff grids but unstable in weak grids, as noted in [18], [19], and [20].   Fig. 8 shows a Bode plot with various power measurement delay parameters. Power measurement delay affects both the gain and phase delay of the system, with the effect on phase delay being dominant. As shown in Table 5, GFM-VSG is unstable when the delay exceeds 0.01. GFM-Droop is stable throughout all the frequency ranges, but the phase margin   becomes less than 30 • , which is regarded as a minimum value for practically maintaining system performance [26].   other hand, the damping factor (D) of GFM-VSG does not affect gain, but a lower D results in a significant phase delay and a higher risk of instability.  Table 7 indicates that a larger M value reduces the phase margin of the system. This tendency is equivalent to the time constant of the measurement delay as discussed in Section III-C3. However, a larger M value does not result in a negative phase margin since the larger M decreases the system gain, unlike a larger measurement delay.

D. SECTION SUMMARY
Here, transfer functions were formulated for GFM power synchronization control and system stability evaluation. The key findings in this section are as follows.  respectively. Thus, GFM-VSG shows a higher risk of causing instability due to the phase delay.
• Hurwitz's stability criterion can discriminate system stability under specific grid conditions and control parameters.
• The Bode plot of the open-loop transfer function ( (11) and (12)) can quantitatively evaluate the effect of grid conditions and GFM parameters on system stability.
• In both GFM-Droop and GFM-VSG, a higher SCR results in higher system gain and risk of system instability.
• In both GFM-Droop and GFM-VSG, a longer power measurement delay results in longer system phase delay and a higher risk of system instability.
• In GFM-Droop, a higher droop gain (K ) results in higher system gain and a higher risk of system instability.
• In GFM-VSG, a lower damping factor (D) results in longer system phase delay and a higher risk of system instability.
• In GFM-VSG, a larger inertia constant (M ) results in lower system gain and longer system phase delay.

IV. SENSITIVITY AGAINST GRID FREQUENCY DEVIATION A. SENSITIVITY FUNCTION FORMULATION
Aside from the transfer function analysis conducted in Section III, this section introduces a method to evaluate the performance of GFM against grid frequency deviation. Fig. 11 shows the systems studied here. To evaluate how much active power GFM injects in response to grid frequency deviation, the input and output of the sensitivity functions [27] are set as the grid frequency deviation ( f g ) and active power output of GFM ( P out ), respectively. O PV (s), respectively. Thus, it is clear that the stability criteria against voltage phase jump and grid frequency deviation are the same as those discussed in Section III.

2) SENSITIVITY FUNCTION FORMULATION
To evaluate the active power output characteristics of GFM against grid frequency deviation, [28], [29], [30] deployed a network frequency perturbation (NFP) plot. The NFP plot is equivalent to the Bode plot of the closed-loop transfer functions, which contain the grid frequency deviation ( f g ) and active power output of GFM ( P out ) as the input and output terms, respectively. The sensitivity functions of GFM-Droop and GFM-VSG, which are derived from Fig. 11, are shown in (19) and (20), respectively.
The system gain in the steady state, which corresponds to the gain when RoCoF is zero, can be derived by setting s as zero (s = 0) in (19) and (20).
According to (21), the active power output of GFM-Droop against grid frequency deviation in steady state is proportional to the inverse of Droop gain (K ). However, the active power output of GFM-VSG against grid frequency deviation in a steady state is proportional to the damping factor (D).

B. NFP PLOT
Here, NFP plot analyses on (19) and (20) are performed to evaluate the active power output characteristics of GFM against grid frequency deviation. The standard parameters here are shown in Table 8. The damping factor of GFM-VSG (D) is tuned so that the power synchronization control openloop (12) shows a phase margin larger than 30 • as per the conditions shown in Table 8. Fig. 12 shows the NFP plot described as per the conditions shown in Table 8. Since the horizontal axis of the NFP plot shows the rate of change of f g , it can be regarded as an indicator similar to RoCoF. In GFM-Droop and GFM-VSG, the phase delay of the active power output in the steady state 42152 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   ( f g = 0) is −180 • . This indicates that the GFM injects active power when the grid frequency drops and vice versa.

1) COMPARISON OF DROOP AND VSG
GFM-VSG shows a higher peak gain than GFM-Droop, although the gains in steady state ( f g = 0) are the same. In addition, GFM-VSG tends to show a phase delay smaller than −180 • in the low-frequency range. This result indicates that GFM-VSG responds to the RoCoF and works faster than the grid frequency deviation, provided that the RoCoF is not excessively fast. Fig. 13 shows the NFP plot with various SCR conditions. In both GFM-Droop and GFM-VSG, a higher SCR results in higher gain and smaller phase delay. Thus, GFMs connected to the stiffer grid contribute to maintaining grid frequency by more quickly injecting highly active power. The difference between GFM-Droop and GFM-VSG in various SCRs is that the highest gain of GFM-VSG tends to be subject to SCR even though GFM-Droop and GFM-VSG show the same gain in the steady state ( f g = 0).   phase delay in the low-frequency range. A significant observation in Fig. 14 is that the characteristics of GFM-Droop with a longer measurement delay (T m = 0.05) are similar to those of the GFM-VSG with no measurement delay (T m = 0) because the measurement delay behaves as synthetic inertia, as denoted in [11]. Fig. 15 shows the NFP plot for various K and D values. In both GFM-Droop and GFM-VSG, the gain in a steady state differs according to (21). In GFM-VSG (D = 50), the peak gain reaches almost ten times that of the steady state gain, although the phase margin of its open-loop transfer function (12) is more than 15 • . This result indicates that a lower D value tends to induce a higher possibility of the inverter reaching its current limitation.

C. SECTION SUMMARY
Here, sensitivity function analysis of the active power output of GFM against grid frequency deviation is performed. The key findings in this section are as follows.
• The stability criterion of the sensitivity function on the active power output characteristics of the GFM against grid frequency deviation is the same as the stability criterion referred to in Section III since both can be represented by the same open-loop transfer function.
• The Bode plot of the closed-loop transfer function ((19) and (20)) can quantitatively evaluate the effect of grid conditions and GFM parameters on the active power output characteristics of GFM against grid frequency deviation.
• GFM-VSG tends to respond faster and inject more active power than GFM-Droop under low RoCoF conditions, such as 1.0 Hz/s.
• In GFM-VSG, the highest gain tends to be subject to SCR.
• In GFM-VSG, the peak gain tends to be high even if the system satisfies the stability criteria.

V. GFM POWER SYNCHRONIZATION CONTROLLER DESIGN PROCEDURE
The GFM power synchronization controller for a stiff grid connection (SCR more than 5) can be designed using the following stepwise procedure. A design example will be introduced in Section VI.
A. GFM-DROOP CONTROLLER DESIGN PROCEDURE 1) Clarify X LCL f , T m and overload capacity of GFM according to system specifications. 2) Specify f 0 , a supposed maximum frequency deviation ( f max ) and supposed minimum X g according to the grid condition. If the minimum X g is unknown, set X g to zero. Suppose  (11) with the parameters determined in steps 1 to 3. Confirm that the phase margin is more than 30 • . If the phase margin is less than 30 • , decrease the K value until the phase margin becomes more than 30 • . 5) Derive the NFP plot of the sensitivity function (19) with the parameters determined in steps 1 to 4. Confirm that the supposed maximum active power output calculated by the maximum gain of the NFP plot and f max ) is less than the overload capacity of the GFM.  (12) with the parameters determined in steps 1 to 3. Confirm that the phase margin is more than 30 • . If the phase margin is less than 30 • , decrease the M value or increase the D value until the phase margin becomes more than 30 • . 5) Derive the NFP plot of the sensitivity function (20) with the parameters determined in steps 1 to 4. Confirm that the supposed maximum active power output calculated by the maximum gain of the NFP plot and f max ) is less than the overload capacity of the GFM. If the supposed maximum active power output exceeds the overload capacity of GFM, adjust the M value or the D value until the supposed maximum active power output is less than the overload. It should be noted that excessively high D values result in higher steady-state gain, as discussed in Section IV-A2.

VI. EXPERIMENTAL RESULTS
To verify the validity of the analysis in Sections III and IV, CHIL experiments [31] are conducted using a real-time simulator and a physical microcontroller equipped with a GFM controller. The CHIL experiments enable us to consider the delay time due to analog input signal conversion, computational execution, and PWM modulation, which are difficult to emulate precisely in transfer function analysis.  Table 9 according to the procedure described in Section V.

B. CONTROLLER DESIGN EXAMPLE 1) POWER SYNCHRONIZATION CONTROLLER OF GFM-DROOP
This section explains the results of the GFM-Droop controller design, following the procedure introduced in Section V-A.
1) X LCL f , T m and overload capacity of GFM are clarified as shown in Table 9. 2) f 0 , f max and supposed minimum X g are clarified as shown in Table 9. 3) Power-frequency droop characteristics are defined as 1.0 pu active power output according to 0.5 Hz frequency deviation. Thus, the K value is determined to be 0.01 by (21). 4) The Bode plot of the open-loop transfer function of the power synchronization controller (11) with the parameters determined in steps 1 to 3 is shown in the solid green line in Fig. 9. The phase margin of this system is 78.406, as shown in Table 6. 5) The NFP plot of the sensitivity function (19) with the parameters determined in steps 1 to 4 is shown in the solid green line Fig. 15. The supposed maximum active power output against f max is 1.0 pu, which is less than the overload capacity of GFM.

2) POWER SYNCHRONIZATION CONTROLLER OF GFM-VSG
This section explains the GFM-VSG controller design example, following the procedure introduced in Section V-B.
1) X LCL f , T m and overload capacity of GFM are clarified as shown in Table 9. 2) f 0 , f max and supposed minimum X g are clarified as shown in Table 9. 3) Power-frequency droop characteristics are defined as 1.0 pu active power output according to 0.5 Hz frequency deviation. Thus, the D value is determined to be 100 by (21). 4) The Bode plot of the open loop transfer function of the power synchronization controller (12) with the parameters determined in steps 1 to 3 is shown in the dashed green line in Fig. 9. The phase margin of this system is 38.829, as shown in Table 6. 5) The NFP plot of the sensitivity function (20) with the parameters determined in steps 1 to 4 is shown in the dashed green line in Fig. 15. The supposed maximum active power output against f max is 2.0 pu, which is less than the overload capacity of GFM.

3) REACTIVE POWER CONTROLLER
The reactive power controller of GFM is designed by deploying the assumption suggested in Section II-C. The Bode plot of the closed-loop reactive power control (G Q (s)/(1 + G Q (s)H (s))) is shown in Fig. 18, which is derived from the parameters shown in Table 9. Finally, the reactive power controller gains are determined to be K Q p = 0.10 and K Q i = 10.00, which realize a closed-loop gain of less than 1.0 across the entire frequency range. C. ACTIVE POWER STEP RESPONSE 1) RESPONSE AGAINST ACTIVE POWER STEP COMMAND Fig. 19 shows the active power output response of GFMs against the active power step command. The responses of both GFM-Droop and GFM-VSG are stable since their controllers are tuned to contain sufficient phase margin. The GFM-VSG shows an overshoot of up to 0.52 pu due to the phase delay, while the GFM-Droop shows no overshoot. This overshoot can be eliminated by increasing the value of D and reducing the phase delay of the system. Fig. 20 shows the active power output response of GFMs against grid frequency sweep in various RoCoF conditions. The response of both GFM-Droop and GFM-VSG is stable, and the response in RoCoF 0.5 Hz/s is mostly similar. However, the GFM-VSG shows an overshoot up to 0.48 pu and responds faster than the GFM-Droop in RoCoF 4,0 Hz/s, as indicated in Section IV-B1. Thus, the GFM-VSG requires a higher overload capacity than GFM-Droop provided that GFM-VSG and GFM-Droop have the same droop characteristics in a steady state, as shown in (21). Further study is needed to evaluate the effect of the overshoot and faster  response of GFM-VSG on the small signal and transient stability of the power system.

D. ANTI-PATTERNS OF POWER SYNCHRONIZATION CONTROLLER DESIGN
This section shows the CHIL experiment results, which demonstrate the instability phenomena indicated by the transfer function analysis conducted in Sections III and IV.

1) UNDERESTIMATING SCR
If the maximum SCR value assumed in the controller design is less than the true SCR value of the grid, there is a risk of instability, as indicated in Section III-C2. Fig. 21 shows the active power output response of the GFM against grid SCR   change for the parameters K = 0.04, D = 25. As indicated in Fig. 7 and Table 4, the active power output from GFM-VSG became unstable when the grid shows a lower reactance value (SCR = 20). While the oscillation occurred with SCR = 20, the stability condition converged when SCR was reduced to 10. Thus, carrying out the power synchronization controller design consistent with the assumed maximum SCR value is essential to avoid the power synchronization control instability of GFM.

2) UNDERESTIMATING MEASUREMENT DELAY
If the measurement delay assumed on the controller design is less than the actual value, there is a risk of instability, as indicated in Section III-C3. Fig. 22 shows the active power output response of GFMs against T m change for the parameters K = 0.04, D = 25. As indicated in Fig. 8 and Table 5, the active power output from GFM-VSG became unstable under larger T m . Thus, carrying out the power synchronization controller design with the consideration of the actual measurement delay is essential to avoid power synchronization control instability of GFM.

3) LESS CONSIDERATION OF DAMPING FACTOR SELECTION
If the D value of GFM-VSG is selected only by simulations over limited conditions, there is a risk of experiencing unexpected instability, as indicated in Section III-C4. Fig. 23 shows the active power output response of GFMs against changing K , D. As indicated in Fig. 9 and Table 6, the active power output from GFM-VSG became unstable when GFM-VSG has a lower damping factor. Thus, the selection of the damping factor of GFM-VSG should be carefully executed, such as by following the procedure proposed in Section V.

VII. CONCLUSION
This paper quantitatively discusses the mechanisms of system instability for GFM power synchronization control of stiff grids by analyzing the phasor-based transfer function of GFM-Droop and GFM-VSG. The key findings of this paper are summarized as follows.
• Systems with power synchronization control by GFM-Droop and GFM-VSG can be represented as secondand third-order systems, respectively. Thus, GFM-VSG presents a higher risk of causing instability due to the phase delay.
• The Bode plot of the phasor-based open-loop transfer function of the power synchronization control can quantitively evaluate the effect of grid conditions and GFM parameters on the system stability.
• The stability criterion of the sensitivity function on the active power output characteristics of the GFM against grid frequency deviation is the same as the stability criterion referred to in Section III since both can be represented by the same open-loop transfer function.
• NFP plots can quantitatively evaluate the effect of grid conditions and GFM parameters on the active power output characteristics of GFM against grid frequency deviation.
• Since a higher SCR results in higher system gain, it causes a higher risk of system instability.
• Factors causing a larger phase delay of the system, such as a longer power measurement delay and lower controller damping, degrade the system stability.
• GFM-VSG tends to respond faster and inject more active power than GFM-Droop under low RoCoF conditions.
Further research should be conducted to investigate the following considerations.
• Simpler formulation and analysis of the system with low SCR in which power synchronization control and reactive power control cannot be easily decoupled • Formulation and analysis of a system that considers the nonlinear characteristics of the power system and GFM, such as the current limitations of GFM • Evaluating the effect of overshoot and faster response of GFM-VSG on small signal and transient stability of the power system